These lecture notes for the first week of https://paradigms.oregonstate.edu/courses/ph441 include a couple of small group activities in which students work with the Gibbs formulation of the entropy.

These lecture notes for the second week of https://paradigms.oregonstate.edu/courses/ph441 involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.

These notes, from the third week of https://paradigms.oregonstate.edu/courses/ph441 cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.

These notes from the fourth week of https://paradigms.oregonstate.edu/courses/ph441 cover blackbody radiation and the Planck distribution. They include a number of small group activities.

1. Set-Up a Sequential Measurement

   1. Add an analyzer to the experiment by:

      1. Break the links between the analyzer and the counters by clicking on the boxes with up and down arrow labels on the analyzer.
      2. Click and drag a new connection from the analyzer to empty space to create a new element. A new analyzer is one of the options.

   2. Measure \(S_z\) twice in succession.

      

      What is the probability that a particle leaving the first analyzer with \(S_z=\frac{+\hbar}{2}\) will be measured by the second analyzer to have \(S_z=\frac{-\hbar}{2}\)?

   3. Set the first analyzer to measure \(S_z\) and the second analyzer to measure \(S_x\).

      

      What have you learned from these experiments?

   
   

2. Try All Combinations of Sequential Measurements

   In the table, enter the probability of a particle exiting the 2nd analyzer with the spin indicated in row if the particle enters the 2nd analyzer with the spin indicated in each column.

   

   
   

3. You can rotate the Stern-Gerlach analyzers to any direction you want (using spherical coordinates).

   Choose an arbitrary direction (not along one of the coordinate axes) for the 1st analyzer and measure the spin along the coordinate directions for the 2nd analyzer.

   

This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.